by Mark Nowitzky,
8/31/96 (revised 2/3/01)
Here’s a too-brief definition of terms:
The musical tuning system in common use in the last couple
hundred years is known as Equal Temperament. It consists of
twelve equally-spaced notes per octave (each a “half-step” apart).
(The math behind it involves the “twelfth-root-of-two”, because a
note’s frequency multiplied by that number twelve times yields a
note having double the original frequency, which is an octave above.)
This system is sometimes referred to as “12tET” (12 tone Equal
Although many people may debate it, Just Intonation is a
more mathematically correct and natural tuning system. Notes in this
system are spaced such that the ratio of their frequencies are equal
to simple integer ratios. There are an infinite number of notes in
an octave, but only a few would be used in a given musical selection.
The 5-Limit Just Intonation tuning system is confined to ratios of
integers which are products of 2, 3, and 5 (prime numbers up to 5).
Increasing the limit to 7 would include the 7th harmonic, which sounds
flat to Western ears.
Okay, skip the math… Here are examples you can listen to:
(Note: The tuning examples use MIDI “pitch wheel” bending to
fine tune the notes. You can check your computer’s sound equipment with the
Pitch Wheel Test before trying the examples.)
The following is the first few bars of the “Promenade” from
Modest Petrovich Moussorgsky’s “Pictures at an Exhibition” (MIDI format),
in equal temperament, and
in just intonation (5-limit).
If you’re still having trouble hearing the differences,
don’t “fret” — try ET and JI side-by-side.
Email me to let me
know which you like better, and why. Thanks!
For more fun with “microtonal” tuning:
- More of my writings on tuning:
- Contrast ET and JI
- “The” Dominant Seventh Chord (revised 2/1/00)
(find out why the word “The” is in quotes!)
- Analysis of Vicentino’s Madonna (revised 1/8/00)
- Some Just Intonation Math
(C is not B#; it’s even worse than that: D is not D)
- Calculate Frequency of Trombone Note (revised 1/18/99)
- Interactive JI Guitar Analyzer (revised 2/8/00)
- Nowitzkian Note Names (revised 5/17/00)
- Is There Always an Optimal Note? (added 6/21/00)
- Join the Alternate Tuning Internet Mailing List!
Dynamic Intonation for Synthesizer Performance,
a thesis by Ben Denckla.
- Kami Rousseau discusses
7-Limit Just Intonation and other tuning systems.
- John Starrett has an excellent collection of
Microtonal Music web resources.
You may email comments to
(I added this counter, ‘cuz lately the topic of Just Intonation is
really taking off. You’re visitor number